In the case of the Boltzmann equation for binary elastic particle interactions, with hard sphere potentials, in three dimensions with a constant angular transitions, the first were developed in [1]}, who showed that it was possible to prove that the summability of moments was true allowing to conclude that the probability density would be . These results were extended seven years later in [2]} for the hard sphere case for binary inelastic collisions with bounded angular transitions, and in [3]}, to show, in the classical case of binary elastic interactions for variable hard potentials, the solutions are exponential weighted in \(L^1({\mathbb {R}}^d)\)  if initially does as well, with a new slower rate need to be strictly positive and finite.
